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Lecture 5 Von Neumann stability analysisMathématiques appliquées (MATH0504-1)B. Dewals, C. Geuzaine

15/12/2020

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Learning objectives of this lecture

Generalize the procedure used in the last lecture to determine the stability of finite difference schemes applied to the diffusion equation

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Outline

Error modes and Fourier decomposition

Von Neuman stability analysis

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1 – Error modes and Fourier decomposition

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Introduction

We consider a linear PDE whose solution is approximated with a time-stepping finite difference scheme.The discretizations in space and time are assumed to be on regular grids xj (with index j), and tn (with index n), respectively, with

The approximation of the solution u on this grid is denoted by

u(x j, tn)⇠ unj

x j = jDx, tn = nDt

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Evolution of the error

Let us consider that at time step 0, an error

appears in the solution, and let us assume that this error (function of the spatial variable x) is both integrable and square integrable.Since the PDE is linear, the time evolution of the initial error will be the same as the one of the solution for the homogeneous problem.

e(x, t0)

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Example

For example, consider the transport equation

with the finite difference approximation

which leads to the difference equation:

ut +aux = f

ut ⇡un+1

j �unj

Dt, ux ⇡

unj+1 �un

j

Dx

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Example

At step n+1 the error satisfies

i.e.

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Stability

The finite difference scheme is said to be stable if the L2-norm (in space) of the error does not diverge with time, i.e. if there exists a real constant C such that

limn!•

||e(x, tn)||2 C||e(x, t0)||2

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Sufficient condition

Using the spatial Fourier decomposition of the error (it exists because the error is integrable)

a sufficient condition for stability is that each mode does not diverge, i.e. that there exists a real constant C such that

limn!•

|e(k, tn)|2 C|e(k, t0)|2, 8k

e(x, tn) =Z

e(k, tn)eikxdk

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Sufficient condition

Indeed, using Parseval’s identity:

(We assume uniform convergence with n of the suite so we can pass the limit under the integration sign.)

limn!•

||e(x, tn)||2 = limn!•

Z|e(k, tn)|2dk

CZ

|e(k, t0)|2dk

=C||e(x, t0)||2

e(k, tn)

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2 – Von Neumann stability analysis

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Stability analysis procedure

By linearity of the equation, each error mode sampled on the spatial grid, defined as

can be studied individually. The procedure is thus:1. Inject in the difference equation2. Simplify the exponentials, which leads to a

(linear) recurrence relation for (usually simply denoted by ) with coefficients that depend on k

3. The scheme is stable if no mode diverges

ek(x j, tn)

e(k, tn)en

ek(x j, tn) = e(k, tn)eikx j

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Take-home message

This Von Neumann stability analysis generalizes the procedure that we applied in the last lecture for the diffusion equation. The solution of the recurrence relation for will involve the amplification factors: if the modulus of any of these amplification factors is greater than one, the scheme diverges.Next, we will apply this general stability analysis to the wave equation.

e(k, tn)